Conditions for existance of definite integral

[mnb96]

Hello,
what are the conditions that a continuous even function f must satisfy in order to have [tex]\int_{-\infty}^{+\infty}f(x)dx < \infty[/tex] ?

 

[tauon]

well, use the definition of the improper integral. the answer to your question is more than obvious if you follow the definition and proprieties of even continuous functions.

tip: only one "part" needs to converge to a finite value.

 

[mnb96]

I guess part of the answer would be that:

the limit for [itex]a\rightarrow +\infty [/itex] of the quantity [itex]F(a)-F(0)[/itex] must be finite.
Here, F denotes the antiderivative of f. Note that now I temporarily assumed that f has an antiderivative, which is not true in general.

- What about functions that don't have an antiderivative but have a proper integral like simple Gaussian functions? Can we state conditions under which such functions surely have a finite integral?

 

[tauon]

What about functions that don't have an antiderivative but have a proper integral like simple Gaussian functions? Can we state conditions under which such functions surely have a finite integral?

You're asking a fairly general question but expect a specific answer. :)

There are a few tricks we can use to test if an improper integral converges but most of them have certain restrictions... so yes we can state conditions under which certain functions have an improper integral, but at the cost of generality.

The most general criterion for the convergence of an improper integral is the Cauchy one... but being general it's also relatively hard to verify.

In the end you have to chose what's more "convenient" .

 

[mnb96]

Thanks. Your answer was useful.
Could you please just point out two more things?

1) what is the Cauchy criterion for convergence of an integral? is it http://en.wikipedia.org/wiki/Integral_test_for_convergence#Statement_of_the_test"?

2) Can we say something more if f is even and we add the restriction that [itex]f(x)>0[/itex] for any x?

 

[tauon]

it seems the forum is screwing my text code.
but no that's not the Cauchy criterion you need (I'll try and write the correct one again, maybe it'll work this time).

as for the second question, yes there are a few results you can use if the function has only positive values.

I'll try to post everything in my next post (the forum is eating or breaking my tex now).

 

[tauon]

the Cauchy criterion for the convergence of an improper integral:

for [tex]f[/tex] : [a, [tex]\infty[/tex] ) [tex]\longrightarrow \mathbb{R}[/tex] , [tex]a \in \mathbb{R}[/tex]
that is integrable on [a, c] , for any c > a
[tex]\int_{a}^{\infty}f[/tex]
exists if and only if for any positive [tex]\epsilon \in \mathbb{R}[/tex]
[tex]\exists K_{\epsilon} \in \mathbb{R}[/tex] such that
[tex]| \int_{c}^{b}f | < \epsilon[/tex]
for any b,
b >= c >= [tex]K_{\epsilon}[/tex]

 

[tauon]

if your function has only positive values
[tex]\int_{a}^{\infty}f[/tex]
converges if and only if the set
[tex]H = \{ I_c | \ I_c = \int_{a}^{c} f , c > a \}[/tex]
is bounded.
in this case
[tex]\int_{a}^{\infty}f = sup H[/tex]

another criterion: if there is another positive function g, such that [tex]f(x} <= g(x)[/tex] for any x > a,
if the infinite integral of g converges the integral of f also converges and is between 0 and the integral of g.

sorry for the late reply

 

[mnb96]

Hi,
thanks a lot for your answers.
That was pretty much what I was looking for.
If you have time, could you please point out some reference to those results you wrote? (title of a book, internet page, or anything...)
Thanks.

 

[tauon]

Hello,

All the results are standard stuff and you can find them in most real analysis textbooks.
A specific example... well you can find them in Robert G. Bartle's, https://www.amazon.com/dp/047105464X/?tag=pfamazon01-20.
Sorry, I don't know any online material for them but I think you can find what you need at a library.

Glad I could help and sorry I answered late. :)